## Abstract

We construct virtual fundamental classes for dg-manifolds whose tangent sheaves have cohomology only in degrees 0 and 1. This condition is analogous to the existence of a perfect obstruction theory in the approach of Behrend and Fantechi fInvent. Math 128 (1997) 45-88] or Li and Tian uJ. Amer. Math. Soc. 11 (1998) 119-174]. Our class is initially defined in K-theory as the class of the structure sheaf of the dg-manifold. We compare our construction with that of Behrend and Fantechi as well as with the original proposal of Kontsevich. We prove a Riemann-Roch type result for dg-manifolds which involves integration over the virtual class. We prove a localization theorem for our virtual classes. We also associate to any dg-manifold of our type a cobordism class of almost complex (smooth) manifolds. This supports the intuition that working with dg-manifolds is the correct algebra-geometric replacement of the analytic technique of"deforming to transversal intersection".

Original language | English (US) |
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Pages (from-to) | 1779-1804 |

Number of pages | 26 |

Journal | Geometry and Topology |

Volume | 13 |

Issue number | 3 |

DOIs | |

State | Published - 2009 |

## Keywords

- Cobordism
- Dgmanifold
- Virtual class